In this explainer, we will learn how to describe atmospheric pressure using various units, including the height of a mercury column.

When looking at a column of fluid, whether a gas or a liquid, the pressure can be calculated as follows.

### Equation: Pressure in a Column of Fluid

The equation used to describe the pressure exerted by the weight of a gas or liquid in a column is where is the pressure, is the density of the fluid, is the acceleration due to gravity (9.81 m/s^{2} for Earth), and is the height above the point we are looking at.

A column of water is usually a tube full of water, and the pressure it exerts on a point can be seen to increase when traveling deeper, since height is larger. This same pressure also exists for the columns of atmosphere, with the top of the column being placed at the very top of the atmosphere of Earth.

Looking at the diagram above, when comparing the pressure of the same heights of water and atmosphere in a column (comparing yellow points at the bottom), the water exerts a higher pressure because it has a higher density.

The atmosphere still applies a pressure, which was first measured by the Italian scientist Torricelli. This was done by placing an inverted tube of mercury into a pool of mercury, then letting the tube drain into the pool, as shown below.

All the way from the top of the atmosphere down to the apparatus exerts a pressure onto the surface of the mercury, which pushes it up the tube, while the mercury in the tube is pulled down by gravity. The forces balance out to make the mercury in the tube fall, but not all the way.

For a 1-metre-long tube at sea level, the distance between the top of the pool of mercury and the top of the mercury in the tube is 760 mm.

The atmospheric pressure changes based on elevation. Being higher up means less pressure, as there is less air.

Letβs look at some examples.

### Example 1: Region Left after Mercury Drop

The apparatus shown in the diagram is used to measure atmospheric pressure. Which of the following occupies region A of the test tube?

- Mercury vapor
- Vacuum
- Glass
- Air

### Answer

When the mercury tube is placed into the pool of mercury, it is completely full of mercury. It is still full of mercury when the bottom is opened, releasing it into the pool, meaning there is nothing to replace the space created in region A.

While there may be small amounts of residual mercury, region A is predominantly a vacuum. It was a full space, and now it is not. Torricelliβs experiment was the earliest artificial vacuum.

The correct answer is B, vacuum.

### Example 2: Atmospheric Pressure at Different Elevations

The apparatus shown in the diagram is used to measure atmospheric pressure. In which case is the apparatus at the greatest height above sea level?

- I
- II
- III
- There is no difference in the apparatusβs height above sea level in the three cases.

### Answer

A greater atmospheric pressure means a higher force pressing down on the mercury pool surface. This would result in pushing the mercury in the tube up higher, so apparatus I has the highest atmospheric pressure, and III has the lowest.

Letβs look at the equation for pressure:

Pressure is related to the density of the fluid, gravity, and height above a particular point. An increase in elevation would correspond to a decrease in the height as you get closer to the edge of the atmosphere. So higher elevations mean lower atmospheric pressures.

The apparatus with the greatest height above sea level would thus be the one with the least atmospheric pressure pressing down on it, which is apparatus III. The answer is thus C.

Atmospheric pressure has several ways it can be expressed in units, as seen in the table below.

Standard Atmosphere | Pascal | Bar | Torr | Millimeters of Mercury |
---|---|---|---|---|

1 atm | 101βββ325 Pa | 1.01 bar | 760 torr | 760 mmHg |

Pascal (Pa) is the SI unit, with 1 pascal being equal to 1 newton per square metre:

The atmosphere of Earth, called the standard atmosphere (atm), is also a unit. The atmospheric pressure of Earth is 1 atm. One atm is 101βββ325 Pa, often shortened to 101 kPa:

Another unit is used for even further shortening, the bar. 1 bar (also abbreviated as βbarβ) is equal to 100βββ000 Pa, so it is often used to shorten pascals. One atm is thus 1.01 bar:

The last two units of pressure, torr and millimetres of mercury, have a 1 to 1 ratio. One torr is one millimetre of mercury, and 760 torr is one atm:

Letβs look at some examples.

### Example 3: Bar to Pascal Conversion

A bar is a unit that is defined as being equal to Pa. Convert a pressure of 0.48 bar to a pressure in pascals.

### Answer

We know that

Or, in other words, there are 100βββ000 pascals per 1 bar:

So if there is only 0.48 bar, we just need to multiply it with this relation:

The units of bar cancel, leaving just Pa. We thus have

So, 0.48 bar is 48βββ000 pascals, equivalent to .

The correct answer is thus E.

### Example 4: Torr Conversions

A torr is a unit that is defined as being equal to the pressure produced by 1 mmHg.

- Convert a pressure of 455 torr to a pressure in pascals.
- Convert a pressure of 455 torr to a pressure in bar.
- 0.606 bar
- 4.55 bar
- 6.37 bar
- 0.637 bar
- 0.619 bar

### Answer

**Part 1**

To convert torr to pascal, letβs relate them both to the standard atmosphere, atm:

We can then relate the conversions as follows:

Or, in other words, for every 101βββ325 pascals, there are 760 torr:

If we multiply this relation by 455 torr, we can cancel the units:

Working out the numbers gives

In scientific notation, 60βββ661 Pa becomes . The answer is thus E.

**Part 2**

The subsequent conversion to bar is easy now that we have it in pascals. 1 bar is 100βββ000 Pa:

So, for every 1 bar, there is 100βββ000 pascals:

Letβs multiply this relation by the pressure in pascals, 60βββ661 Pa:

Units of pascal cancel, giving which means 455 torr in bar is 0.606 bar. The answer is thus A.

Now that we have seen how to convert between units of atmospheric pressure, letβs look back at the original equation for calculating pressure:

There are many different units of pressure, and those units can have prefixes that change them further, such as using kilopascal (kPa), or measuring using centimetre of mercury (cmHg), instead of millimetres of mercury (mmHg). Most times, the final result will be in pascals when using this equation, but always be wary of the units used.

Letβs look at an example.

### Example 5: Finding the Upward Pressure on a Mercury Column

The apparatus shown in the diagram is used to measure atmospheric pressure. Find the upward pressure on the mercury column. Use a value of 13βββ595 kg/m^{3} for the density of mercury.

- 133 kPa
- 175 kPa
- 203 kPa
- 50.6 kPa
- 101 kPa

### Answer

The upward pressure inside the column of mercury depends on its height above the pool, which is thankfully given, 0.76 m. In the equation for pressure, this is the value of :

The force of gravity on Earth, , is 9.81 m/s^{2}. We are also given the density , which is 13βββ595 kg/m^{3}. We thus have all the variables we need to fit into the equation for pressure:

The units of metres in gravity and height cancel with the ones given by density. Multiplying everything through gives

Pascals are equivalent to newtons per square metre (N/m^{2}) and newtons are equivalent to kilogram-metres per second squared. We can convert the units we have to newtons per square metre as follows:

Thus, the final pressure is

101βββ358 Pa is 101 kPa. The correct answer is thus E.

Letβs summarize what we have learned in this explainer.

### Key Points

- The equation to describe the pressure of a liquid or gas is where is the pressure, is the density of the fluid, is the force due to gravity, and is the height.
- A column of mercury can be used to measure atmospheric pressure.
- 1 atmΒ Β 101βββ325 PaΒ Β 1.01 bar
- 1 atmΒ Β 760 mmHgΒ Β 760 torr